# Fall 2013

## Friday, September 20, 2013, 1:30-2:30pm; Université de Montreal, Pavillon André-Aisenstadt, Room: 4336.      Huaihui Chen, Nanjing Normal University.

### Title:

The Schwarz-Pick Lemma for real planar harmonic mappings.

## Monday, September 23, 2013, 1:30-2:30pm; Université de Montreal, Pavillon André-Aisenstadt, Salle: 5448.      Alexander Strohmaier, Loughborough University, UK.

### Title:

Ray Splitting Billiards and Quantum Ergodicity.

### Abstract:

The semi-classical limit of smooth and continuous quantum systems is well covered in the literature and leads to classical mechanics, thus establishing Bohr's correspondence principle. The situation changes when one deals with Quantum mechanical systems with discontinuous metrics. Ray splitting occurs and the semi-classical limit is not described by a classical flow any more. I will explain to what extent the semi-group emerging from a ray-splitting flow replaces the classical dynamics in this context. I will show that one can prove Quantum ergodicity in this context under very natural assumptions. (joint work with D. Jakobson and Y. Safarov)

## Friday, October 4, 2013 at 1:30-2:30pm; Concordia University, Conference Room: LB 921-4. Susanna Dann, University of Missouri at Columbia.

### Title:

The Busemann-Petty problem in the complex hyperbolic space.

### Abstract:

The classical Busemann-Petty problem asks whether origin-symmetric convex bodies in R^n with corresponding smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative for n <= 4 and negative for n >= 5. We study this problem in the complex hyperbolic n-space and prove that the answer is affirmative for n <= 2 and negative for n >= 3.

## Friday, October 11, 2013, 1:30-2:30pm; Université de Montreal/CRM, Salle: 4336.      Martin Klimes, DMS, Université de Montreal.

### Title:

Confluence of singularities of ODE via Borel-Laplace transformations.

### Abstract:

We consider the question of existence and domain of bounded solutions of systems of analytic ODEs near a singular point. In case of a multiple singularity, these solutions often possess a divergent asymptotic expansion. A standard method for obtaining such a solution is then to take the Borel-Laplace sum of the divergent series. We explain on an example how to generalize the Borel-Laplace method to study these solutions under small analytic perturbations when a double singularity splits into two simple singularities. The example we consider is that of analytic parameter dependent systems of ODEs $(x^2-\epsilon)\frac{dy}{dx}=My+f(x,y,\epsilon)$, under a generic condition that the matrix $M$ of the linear part is invertible. Our approach allows for a unified treatment in the spirit of resurgent analysis for all values of the perturbation parameter $\epsilon$ in a neighborhood of 0.

## Friday, October 18, 2013, 2:30-3:30pm; Université de Montreal/CRM, Salle: 5340.      Maamoun Turkawi, Montreal.

### Titre:

Espace de Sobolev et Hardy-Sobolev dans le cas des graphes.

### Résumé:

Soit \Gamma un graphe vérifiant des hypothèses géométriques convenables. On va donner diverses caractérisations équivalents pour les espaces de Sobolev et Hardy-Sobolev sur \Gamma, en termes de fonctions maximales, fonctions de type Hajlasz ou décompositions atomiques.

## Friday, October 25, 2013 at 2:30-3:30pm; Université de Montreal/CRM, Salle: 5340. Javad Mashreghi, Université Laval.

### Title:

Composition operators on function spaces.

### Abstract:

Let $\varphi:$\mathbb{D} \to $\mathbb{D}$ be analytic and consider the (composition) mapping $C_\varphi(f) = f \circ \varphi$ on the family of holomorphic functions. Then the Hardy space $H^2$ is closed under all $C_\varphis$’s. This is the classical {\em subordination principle of Littlewood} stated in the language of operator theory. One can ask a similar question for any function space on $\mathbb{D}$. We discuss the composition operators on model spaces $K_\Theta$ and de Branges-Rovnyak spaces $\mathcal{H}(b)$. In both cases, in particular the latter, the question is still wide open and just partial results are available.

## Friday, November 1, 2013 at 1:30-2:30pm; McGill, Room: Burnside 920. Thierry Daude, Université de Cergy-Pontoise.

### Title:

Inverse scattering at fixed energy in black hole spacetimes.

### Abstract:

In this talk, we first describe a class of axisymmetric, electrically charged, spacetimes with positive cosmological constant, called Kerr-Newmann-de-Sitter black holes, which are exact solutions of the Einstein equations. The main question we adress is the following: can we determine the metric of such black holes by observing waves at the "infinities" of the spacetime? Precisely, the considered waves will be massless Dirac fields evolving in the outer region of Kerr-Newman-de-Sitter black holes. We shall define the corresponding scattering matrix, object that encodes the far field behavior of these Dirac fields from the point of view of static observers. We finally shall show that the metric of such black holes is uniquely determined by the knowledge of this scattering matrix at a fixed energy. This result was obtained in collaboration with François Nicoleau (Université de Nantes).

## Friday, November 8, 2013 at 1:30-2:30pm; UdeM, Salle: 4186. NOTE THE CHANGE OF VENUE! Annalisa Panati, McGill University.

### Title:

Infrared (and ultraviolet) aspects of a model of QFT on a static space time.

### Abstract:

We consider the Nelson model with variable coeffcients, which can be seen as a model describing a particle interacting with a scalar field on a static space time. We consider the problem of the existence of the ground state, showing that it depends on the decay rate of the coeffcients at infinity. We also show that it is possible to remove the ultraviolet cutoff, as it is in the flat case. We'll explain some open conjecture. (joint work with C.Gérard, F.Hiroshima, A.Suzuki)

## Friday, November 15, 2013 at 2-3pm; Concordia University, Conference Room: LB 921-4. NOTE THE TIME! Scott Rodney, Cape Breton University.

### Title:

Compact embeddings for generalized Sobolev spaces and applications.

### Abstract:

This talk will focus on functional properties of generalized Sobolev spaces and their connections to degenerate elliptic partial differential equations. To begin, generalized Sobolev spaces, and weighted degenerate Sobolev spaces will be defined and their connections to degenerate PDEs discussed. Following this, a general embedding result associated to such spaces will be presented in the context of important examples. Examples to be discussed involve degeneracies encoded by use of Muckenhoupt $A_p$ weights and also degenerate elliptic PDEs defined with respect to collections of Lipschitz vector fields.

## Friday, November 22, 2013, 2:30-3:30pm; UdeM, Salle: 4186.      Damir Kinzebulatov, Fields Institute.

### Title:

Unique continuation for Schroedinger operators (joint work with Leonid Shartser).

### Abstract:

The property of unique continuation (UC) of solutions of a PDE plays a fundamental role in Analysis, as well as its applications to Mathematical Physics and Geometry (e.g. for proving that Schroedinger operators do not have positive eigenvalues). The UC property is: "if a solution vanishes on an open subset (of its connected domain) then it must be equal to zero everywhere". We prove UC property for solutions of the differential inequality |\Delta u| \leq |Vu| for V from a wide class of potentials (including "locally in L^{n/2}" class) and u in a space of solutions Y_V containing all eigenfunctions of the corresponding self-adjoint Schroedinger operator, extending the classical results by D. Jerison, C. Kenig [Ann. Math 1984] and E. Stein [Ann. Math,1984].

## Friday, November 29, 2013, 1:30-2:30pm; McGill, Room: Burnside 920.      Emily Dryden, Bucknell University.

### Title:

Isospectrality and heat content.

### Abstract:

The examples of isospectral non-isometric drums'' constructed by Carolyn Gordon, David Webb, and Scott Wolpert show that one cannot hear the shape of a piecewise smooth planar domain $D$. They also tell us that the eigenvalues of the Dirichlet Laplace operator acting on smooth functions on $D$ form an incomplete set of geometric invariants, and it is therefore natural to look for ways to distinguish such non-isometric sound-alike drums. We will discuss what we can learn from heating these drums and studying the amount of heat in them over time. This is joint work with Michiel van den Berg and Thomas Kappeler.

## Friday, December 6, 2013, 1:30-2:30pm, UdeM, Salle: 4186.      Alexandre Girouard, Université Laval.

### Title:

Spectral geometry of the Steklov problem: asymptotics and invariants.

### Abstract:

In this talk, I will describe a recent joint work with L. Parnovski, I. Polterovich and D. Sher in which we have obtained precise asymptotics for the Steklov eigenvalues on a compact Riemannian surface with boundary. This has lead to a proof that the number of connected components of the boundary, as well as their lengths, are invariants of the Steklov spectrum. The proofs are based on pseudodifferential techniques for the Dirichlet-to-Neumann operator and on a number-theoretic argument.